Modern Problems of Mathematical Physics and Their Applications

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (30 September 2021) | Viewed by 52396

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Higher Military Aviation School of the Republic of Uzbekistan, Karshi City 180100, Uzbekistan
Interests: differential equations; equations of mathematical physics; partial differential equations; solving ill-posed problems; Cauchy problem for first order elliptic systems; solving integral equations; Cauchy problem for elliptic equations; approximate solutions of the Cauchy problem for elliptic systems and equations

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1. Baikal School of BRICS, Irkutsk National Research Technical University, 664074 Irkutsk, Russia
2. Department of Applied Mathematics and Programming, South Ural State University, Lenin Prospect 76, 454080 Chelyabinsk, Russia
Interests: numerical analysis; solving integral equations; solving ODEs and PDEs; solving ill-posed problems; fuzzy mathematics; stochastic arithmetic; CADNA library; CESTAC method; solving biomathematical models; iterative methods; numerical methods
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Special Issue Information

Dear Colleagues,

We invite you to submit your research papers in the field of modern mathematical physics problems to this Special Issue, entitled “Modern Problems of Mathematical Physics and Their Applications”, of the journal Axioms. We seek studies on new and innovative approaches to solving mathematical physics problems in linear and nonlinear forms. We also aim to cover high-dimensional problems and systems of equations. We welcome submissions presenting new theoretical results, structural investigations, new models and algorithmic approaches, and new applications of mathematical physics problems.

Prof. Dr. Davron Aslonqulovich Juraev
Prof. Dr. Samad Noeiaghdam
Guest Editors

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Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

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Keywords

  • integral equations
  • ill-posed problems
  • ordinary differential equations
  • partial differential equations
  • system of equations
  • fractional problems
  • linear and nonlinear problems
  • fuzzy problems
  • numerical methods
  • analytical methods
  • semianalytical methods
  • convergence analysis
  • error analysis
  • mathematical models

Published Papers (20 papers)

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Editorial

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6 pages, 222 KiB  
Editorial
Modern Problems of Mathematical Physics and Their Applications
by Davron Aslonqulovich Juraev and Samad Noeiaghdam
Axioms 2022, 11(2), 45; https://doi.org/10.3390/axioms11020045 - 23 Jan 2022
Cited by 10 | Viewed by 2418
Abstract
There are many applications of mathematical physics in several fields of basic science and engineering [...] Full article
(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)

Research

Jump to: Editorial, Review

17 pages, 335 KiB  
Article
Oscillatory Behavior of Third-Order Quasi-Linear Neutral Differential Equations
by Belgees Qaraad, Osama Moaaz, Shyam Sundar Santra, Samad Noeiaghdam, Denis Sidorov and Elmetwally M. Elabbasy
Axioms 2021, 10(4), 346; https://doi.org/10.3390/axioms10040346 - 17 Dec 2021
Cited by 3 | Viewed by 1893
Abstract
In this paper, we consider a class of quasilinear third-order differential equations with a delay argument. We establish some conditions of such certain third-order quasi-linear neutral differential equation as oscillatory or almost oscillatory. Those criteria improve, complement and simplify a number of existing [...] Read more.
In this paper, we consider a class of quasilinear third-order differential equations with a delay argument. We establish some conditions of such certain third-order quasi-linear neutral differential equation as oscillatory or almost oscillatory. Those criteria improve, complement and simplify a number of existing results in the literature. Some examples are given to illustrate the importance of our results. Full article
(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)
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19 pages, 706 KiB  
Article
Multi-Asset Barrier Options Pricing by Collocation BEM (with Matlab® Code)
by Alessandra Aimi and Chiara Guardasoni
Axioms 2021, 10(4), 301; https://doi.org/10.3390/axioms10040301 - 12 Nov 2021
Cited by 4 | Viewed by 1862
Abstract
In this paper, we extend the SABO technique (Semi-Analytical method for Barrier Options), based on collocation Boundary Element Method (BEM), to the pricing of Barrier Options with payoff dependent on more than one asset. The efficiency and accuracy already revealed in the case [...] Read more.
In this paper, we extend the SABO technique (Semi-Analytical method for Barrier Options), based on collocation Boundary Element Method (BEM), to the pricing of Barrier Options with payoff dependent on more than one asset. The efficiency and accuracy already revealed in the case of a single asset is confirmed by the presented numerical results. Full article
(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)
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32 pages, 12108 KiB  
Article
Modeling Multi-Dimensional Public Opinion Process Based on Complex Network Dynamics Model in the Context of Derived Topics
by Tinggui Chen, Xiaohua Yin, Jianjun Yang, Guodong Cong and Guoping Li
Axioms 2021, 10(4), 270; https://doi.org/10.3390/axioms10040270 - 21 Oct 2021
Cited by 38 | Viewed by 3578
Abstract
With the rapid development of the Internet, the speed with which information can be updated and propagated has accelerated, resulting in wide variations in public opinion. Usually, after the occurrence of some newsworthy event, discussion topics are generated in networks that influence the [...] Read more.
With the rapid development of the Internet, the speed with which information can be updated and propagated has accelerated, resulting in wide variations in public opinion. Usually, after the occurrence of some newsworthy event, discussion topics are generated in networks that influence the formation of initial public opinion. After a period of propagation, some of these topics are further derived into new subtopics, which intertwine with the initial public opinion to form a multidimensional public opinion. This paper is concerned with the formation process of multi-dimensional public opinion in the context of derived topics. Firstly, the initial public opinion variation mechanism is introduced to reveal the formation process of derived subtopics, then Brownian motion is used to determine the subtopic propagation parameters and their propagation is studied based on complex network dynamics according to the principle of evolution. The formula of basic reproductive number is introduced to determine whether derived subtopics can form derived public opinion, thereby revealing the whole process of multi-dimensional public opinion formation. Secondly, through simulation experiments, the influences of various factors, such as the degree of information alienation, environmental forces, topic correlation coefficients, the amount of information contained in subtopics, and network topology on the formation of multi-dimensional public opinion are studied. The simulation results show that: (1) Environmental forces and the amount of information contained in subtopics are key factors affecting the formation of multi-dimensional public opinion. Among them, environmental forces have a greater impact on the number of subtopics, and the amount of information contained in subtopics determines whether the subtopic can be the key factor that forms the derived public opinion. (2) Only when the degree of information alienation reaches a certain level, will derived subtopics emerge. At the same time, the degree of information alienation has a greater impact on the number of derived subtopics, but it has a small impact on the dimensions of the final public opinion. (3) The network topology does not have much impact on the number of derived subtopics but has a greater impact on the number of individuals participating in the discussion of subtopics. The multidimensional public opinion dimension formed by the network topology with a high aggregation coefficient and small average path length is higher. Finally, a practical case verifies the rationality and effectiveness of the model proposed in this paper. Full article
(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)
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17 pages, 332 KiB  
Article
Existence of Mild Solutions for Multi-Term Time-Fractional Random Integro-Differential Equations with Random Carathéodory Conditions
by Amadou Diop and Wei-Shih Du
Axioms 2021, 10(4), 252; https://doi.org/10.3390/axioms10040252 - 12 Oct 2021
Cited by 2 | Viewed by 1520
Abstract
In this paper, we investigate the existence of mild solutions to a multi-term fractional integro-differential equation with random effects. Our results are mainly relied upon stochastic analysis, Mönch’s fixed point theorem combined with a random fixed point theorem with stochastic domain, measure of [...] Read more.
In this paper, we investigate the existence of mild solutions to a multi-term fractional integro-differential equation with random effects. Our results are mainly relied upon stochastic analysis, Mönch’s fixed point theorem combined with a random fixed point theorem with stochastic domain, measure of noncompactness and resolvent family theory. Under the condition that the nonlinear term is of Carathéodory type and satisfies some weakly compactness condition, we establish the existence of random mild solutions. A nontrivial example illustrating our main result is also given. Full article
(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)
11 pages, 257 KiB  
Article
On Solvability Conditions for a Certain Conjugation Problem
by Vladimir Vasilyev and Nikolai Eberlein
Axioms 2021, 10(3), 234; https://doi.org/10.3390/axioms10030234 - 20 Sep 2021
Cited by 2 | Viewed by 1515
Abstract
We study a certain conjugation problem for a pair of elliptic pseudo-differential equations with homogeneous symbols inside and outside of a plane sector. The solution is sought in corresponding Sobolev–Slobodetskii spaces. Using the wave factorization concept for elliptic symbols, we derive a general [...] Read more.
We study a certain conjugation problem for a pair of elliptic pseudo-differential equations with homogeneous symbols inside and outside of a plane sector. The solution is sought in corresponding Sobolev–Slobodetskii spaces. Using the wave factorization concept for elliptic symbols, we derive a general solution of the conjugation problem. Adding some complementary conditions, we obtain a system of linear integral equations. If the symbols are homogeneous, then we can apply the Mellin transform to such a system to reduce it to a system of linear algebraic equations with respect to unknown functions. Full article
(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)
13 pages, 287 KiB  
Article
Closed-Form Solutions of Linear Ordinary Differential Equations with General Boundary Conditions
by Efthimios Providas, Stefanos Zaoutsos and Ioannis Faraslis
Axioms 2021, 10(3), 226; https://doi.org/10.3390/axioms10030226 - 14 Sep 2021
Cited by 1 | Viewed by 1788
Abstract
This paper deals with the solution of boundary value problems for ordinary differential equations with general boundary conditions. We obtain closed-form solutions in a symbolic form of problems with the general n-th order differential operator, as well as the composition of linear [...] Read more.
This paper deals with the solution of boundary value problems for ordinary differential equations with general boundary conditions. We obtain closed-form solutions in a symbolic form of problems with the general n-th order differential operator, as well as the composition of linear operators. The method is based on the theory of the extensions of linear operators in Banach spaces. Full article
(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)
23 pages, 2374 KiB  
Article
A Dynamic Model of Multiple Time-Delay Interactions between the Virus-Infected Cells and Body’s Immune System with Autoimmune Diseases
by Hoang Pham
Axioms 2021, 10(3), 216; https://doi.org/10.3390/axioms10030216 - 7 Sep 2021
Cited by 5 | Viewed by 2020
Abstract
The immune system is a complex interconnected network consisting of many parts including organs, tissues, cells, molecules and proteins that work together to protect the body from illness when germs enter the body. An autoimmune disease is a disease in which the body’s [...] Read more.
The immune system is a complex interconnected network consisting of many parts including organs, tissues, cells, molecules and proteins that work together to protect the body from illness when germs enter the body. An autoimmune disease is a disease in which the body’s immune system attacks healthy cells. It is known that when the immune system is working properly, it can clearly recognize and kill the abnormal cells and virus-infected cells. But when it doesn’t work properly, the human body will not be able to recognize the virus-infected cells and, therefore, it can attack the body’s healthy cells when there is no invader or does not stop an attack after the invader has been killed, resulting in autoimmune disease.; This paper presents a mathematical modeling of the virus-infected development in the body’s immune system considering the multiple time-delay interactions between the immune cells and virus-infected cells with autoimmune disease. The proposed model aims to determine the dynamic progression of virus-infected cell growth in the immune system. The patterns of how the virus-infected cells spread and the development of the body’s immune cells with respect to time delays will be derived in the form of a system of delay partial differential equations. The model can be used to determine whether the virus-infected free state can be reached or not as time progresses. It also can be used to predict the number of the body’s immune cells at any given time. Several numerical examples are discussed to illustrate the proposed model. The model can provide a real understanding of the transmission dynamics and other significant factors of the virus-infected disease and the body’s immune system subject to the time delay, including approaches to reduce the growth rate of virus-infected cell and the autoimmune disease as well as to enhance the immune effector cells. Full article
(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)
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17 pages, 17014 KiB  
Article
Reinitializing Sea Surface Temperature in the Ensemble Intermediate Coupled Model for Improved Forecasts
by Sittisak Injan, Angkool Wangwongchai, Usa Humphries, Amir Khan and Abdullahi Yusuf
Axioms 2021, 10(3), 189; https://doi.org/10.3390/axioms10030189 - 17 Aug 2021
Cited by 2 | Viewed by 1834
Abstract
The Ensemble Intermediate Coupled Model (EICM) is a model used for studying the El Niño-Southern Oscillation (ENSO) phenomenon in the Pacific Ocean, which is anomalies in the Sea Surface Temperature (SST) are observed. This research aims to implement Cressman to improve SST forecasts. [...] Read more.
The Ensemble Intermediate Coupled Model (EICM) is a model used for studying the El Niño-Southern Oscillation (ENSO) phenomenon in the Pacific Ocean, which is anomalies in the Sea Surface Temperature (SST) are observed. This research aims to implement Cressman to improve SST forecasts. The simulation considers two cases in this work: the control case and the Cressman initialized case. These cases are simulations using different inputs where the two inputs differ in terms of their resolution and data source. The Cressman method is used to initialize the model with an analysis product based on satellite data and in situ data such as ships, buoys, and Argo floats, with a resolution of 0.25 × 0.25 degrees. The results of this inclusion are the Cressman Initialized Ensemble Intermediate Coupled Model (CIEICM). Forecasting of the sea surface temperature anomalies was conducted using both the EICM and the CIEICM. The results show that the calculation of SST field from the CIEICM was more accurate than that from the EICM. The forecast using the CIEICM initialization with the higher-resolution satellite-based analysis at a 6-month lead time improved the root mean square deviation to 0.794 from 0.808 and the correlation coefficient to 0.630 from 0.611, compared the control model that was directly initialized with the low-resolution in-situ-based analysis. Full article
(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)
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11 pages, 298 KiB  
Article
Application of a Generalized Secant Method to Nonlinear Equations with Complex Roots
by Avram Sidi
Axioms 2021, 10(3), 169; https://doi.org/10.3390/axioms10030169 - 29 Jul 2021
Cited by 1 | Viewed by 2466
Abstract
The secant method is a very effective numerical procedure used for solving nonlinear equations of the form f(x)=0. In a recent work (A. Sidi, Generalization of the secant method for nonlinear equations. Appl. Math. E-Notes, 8:115–123, [...] Read more.
The secant method is a very effective numerical procedure used for solving nonlinear equations of the form f(x)=0. In a recent work (A. Sidi, Generalization of the secant method for nonlinear equations. Appl. Math. E-Notes, 8:115–123, 2008), we presented a generalization of the secant method that uses only one evaluation of f(x) per iteration, and we provided a local convergence theory for it that concerns real roots. For each integer k, this method generates a sequence {xn} of approximations to a real root of f(x), where, for nk, xn+1=xnf(xn)/pn,k(xn), pn,k(x) being the polynomial of degree k that interpolates f(x) at xn,xn1,,xnk, the order sk of this method satisfying 1<sk<2. Clearly, when k=1, this method reduces to the secant method with s1=(1+5)/2. In addition, s1<s2<s3<, such that limksk=2. In this note, we study the application of this method to simple complex roots of a function f(z). We show that the local convergence theory developed for real roots can be extended almost as is to complex roots, provided suitable assumptions and justifications are made. We illustrate the theory with two numerical examples. Full article
(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)
20 pages, 339 KiB  
Article
Critical Indices and Self-Similar Power Transform
by Simon Gluzman
Axioms 2021, 10(3), 162; https://doi.org/10.3390/axioms10030162 - 26 Jul 2021
Cited by 5 | Viewed by 3248
Abstract
“Odd” factor approximants of the special form suggested by Gluzman and Yukalov (J. Math. Chem. 2006, 39, 47) are amenable to optimization by power transformation and can be successfully applied to critical phenomena. The approach is based on the idea that the critical [...] Read more.
“Odd” factor approximants of the special form suggested by Gluzman and Yukalov (J. Math. Chem. 2006, 39, 47) are amenable to optimization by power transformation and can be successfully applied to critical phenomena. The approach is based on the idea that the critical index by itself should be optimized through the parameters of power transform to be calculated from the minimal sensitivity (derivative) optimization condition. The critical index is a product of the algebraic self-similar renormalization which contributes to the expressions the set of control parameters typical to the algebraic self-similar renormalization, and of the power transform which corrects them even further. The parameter of power transformation is, in a nutshell, the multiplier connecting the critical exponent and the correction-to-scaling exponent. We mostly study the minimal model of critical phenomena based on expansions with only two coefficients and critical points. The optimization appears to bring quite accurate, uniquely defined results given by simple formulas. Many important cases of critical phenomena are covered by the simple formula. For the longer series, the optimization condition possesses multiple solutions, and additional constraints should be applied. In particular, we constrain the sought solution by requiring it to be the best in prediction of the coefficients not employed in its construction. In principle, the error/measure of such prediction can be optimized by itself, with respect to the parameter of power transform. Methods of calculation based on optimized power-transformed factors are applied and results presented for critical indices of several key models of conductivity and viscosity of random media, swelling of polymers, permeability in two-dimensional channels. Several quantum mechanical problems are discussed as well. Full article
(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)
16 pages, 4368 KiB  
Article
Strong Interacting Internal Waves in Rotating Ocean: Novel Fractional Approach
by Pundikala Veeresha, Haci Mehmet Baskonus and Wei Gao
Axioms 2021, 10(2), 123; https://doi.org/10.3390/axioms10020123 - 16 Jun 2021
Cited by 53 | Viewed by 2394
Abstract
The main objective of the present study is to analyze the nature and capture the corresponding consequences of the solution obtained for the Gardner–Ostrovsky equation with the help of the q-homotopy analysis transform technique (q-HATT). In the rotating ocean, the [...] Read more.
The main objective of the present study is to analyze the nature and capture the corresponding consequences of the solution obtained for the Gardner–Ostrovsky equation with the help of the q-homotopy analysis transform technique (q-HATT). In the rotating ocean, the considered equations exemplify strong interacting internal waves. The fractional operator employed in the present study is used in order to illustrate its importance in generalizing the models associated with kernel singular. The fixed-point theorem and the Banach space are considered to present the existence and uniqueness within the frame of the Caputo–Fabrizio (CF) fractional operator. Furthermore, for different fractional orders, the nature has been captured in plots. The realized consequences confirm that the considered procedure is reliable and highly methodical for investigating the consequences related to the nonlinear models of both integer and fractional order. Full article
(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)
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10 pages, 455 KiB  
Article
Classical Partition Function for Non-Relativistic Gravity
by Mir Hameeda, Angelo Plastino, Mario Carlos Rocca and Javier Zamora
Axioms 2021, 10(2), 121; https://doi.org/10.3390/axioms10020121 - 16 Jun 2021
Cited by 3 | Viewed by 2181
Abstract
We considered the canonical gravitational partition function Z associated to the classical Boltzmann–Gibbs (BG) distribution eβHZ. It is popularly thought that it cannot be built up because the integral involved in constructing Z diverges at the origin. Contrariwise, [...] Read more.
We considered the canonical gravitational partition function Z associated to the classical Boltzmann–Gibbs (BG) distribution eβHZ. It is popularly thought that it cannot be built up because the integral involved in constructing Z diverges at the origin. Contrariwise, it was shown in (Physica A 497 (2018) 310), by appeal to sophisticated mathematics developed in the second half of the last century, that this is not so. Z can indeed be computed by recourse to (A) the analytical extension treatments of Gradshteyn and Rizhik and Guelfand and Shilov, that permit tackling some divergent integrals and (B) the dimensional regularization approach. Only one special instance was discussed in the above reference. In this work, we obtain the classical partition function for Newton’s gravity in the four cases that immediately come to mind. Full article
(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)
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9 pages, 308 KiB  
Article
On a Class of Isoperimetric Constrained Controlled Optimization Problems
by Savin Treanţă
Axioms 2021, 10(2), 112; https://doi.org/10.3390/axioms10020112 - 3 Jun 2021
Cited by 9 | Viewed by 1871
Abstract
In this paper, we investigate the Lagrange dynamics generated by a class of isoperimetric constrained controlled optimization problems involving second-order partial derivatives and boundary conditions. More precisely, we derive necessary optimality conditions for the considered class of variational control problems governed by path-independent [...] Read more.
In this paper, we investigate the Lagrange dynamics generated by a class of isoperimetric constrained controlled optimization problems involving second-order partial derivatives and boundary conditions. More precisely, we derive necessary optimality conditions for the considered class of variational control problems governed by path-independent curvilinear integral functionals. Moreover, the theoretical results presented in the paper are accompanied by an illustrative example. Furthermore, an algorithm is proposed to emphasize the steps to be followed to solve a control problem such as the one studied in this paper. Full article
(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)
16 pages, 2772 KiB  
Article
Monitoring and Recognizing Enterprise Public Opinion from High-Risk Users Based on User Portrait and Random Forest Algorithm
by Tinggui Chen, Xiaohua Yin, Lijuan Peng, Jingtao Rong, Jianjun Yang and Guodong Cong
Axioms 2021, 10(2), 106; https://doi.org/10.3390/axioms10020106 - 27 May 2021
Cited by 44 | Viewed by 5404
Abstract
With the rapid development of “We media” technology, netizens can freely express their opinions regarding enterprise products on a network platform. Consequently, online public opinion about enterprises has become a prominent issue. Negative comments posted by some netizens may trigger negative public opinion, [...] Read more.
With the rapid development of “We media” technology, netizens can freely express their opinions regarding enterprise products on a network platform. Consequently, online public opinion about enterprises has become a prominent issue. Negative comments posted by some netizens may trigger negative public opinion, which can have a significant impact on an enterprise’s image. From the perspective of helping enterprises deal with negative public opinion, this paper combines user portrait technology and a random forest algorithm to help enterprises identify high-risk users who have posted negative comments and thus may trigger negative public opinion. In this way, enterprises can monitor the public opinion of high-risk users to prevent negative public opinion events. Firstly, we crawled the information of users participating in discussions of product experience, and we constructed a portrait of enterprise public opinion users. Then, the characteristics of the portraits were quantified into indicators such as the user’s activity, the user’s influence, and the user’s emotional tendency, and the indicators were sorted. According to the order of the indicators, the users were divided into high-risk, moderate-risk, and low-risk categories. Next, a supervised high-risk user identification model for this classification was established, based on a random forest algorithm. In turn, the trained random forest identifier can be used to predict whether the authors of newly published public opinion information are high-risk users. Finally, a back propagation neural network algorithm was used to identify users and compared with the results of model recognition in this paper. The results showed that the average recognition accuracy of the back propagation neural network is only 72.33%, while the average recognition accuracy of the model constructed in this paper is as high as 98.49%, which verifies the feasibility and accuracy of the proposed random forest recognition method. Full article
(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)
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9 pages, 268 KiB  
Article
Comments on the Navier–Stokes Problem
by Alexander G. Ramm
Axioms 2021, 10(2), 95; https://doi.org/10.3390/axioms10020095 - 20 May 2021
Cited by 3 | Viewed by 1647
Abstract
The aim of this paper is to explain for broad audience the author’s result concerning the Navier–Stokes problem (NSP) in R3 without boundaries. It is proved that the NSP is contradictory in the following sense: if one assumes that the initial data [...] Read more.
The aim of this paper is to explain for broad audience the author’s result concerning the Navier–Stokes problem (NSP) in R3 without boundaries. It is proved that the NSP is contradictory in the following sense: if one assumes that the initial data v(x,0)0, ·v(x,0)=0 and the solution to the NSP exists for all t0, then one proves that the solution v(x,t) to the NSP has the property v(x,0)=0. This paradox shows that the NSP is not a correct description of the fluid mechanics problem and the NSP does not have a solution. In the exceptional case, when the data are equal to zero, the solution v(x,t) to the NSP exists for all t0 and is equal to zero, v(x,t)0. Thus, one of the millennium problems is solved. Full article
(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)
13 pages, 295 KiB  
Article
Degenerate Canonical Forms of Ordinary Second-Order Linear Homogeneous Differential Equations
by Dimitris M. Christodoulou, Eric Kehoe and Qutaibeh D. Katatbeh
Axioms 2021, 10(2), 94; https://doi.org/10.3390/axioms10020094 - 19 May 2021
Cited by 1 | Viewed by 1952
Abstract
For each fundamental and widely used ordinary second-order linear homogeneous differential equation of mathematical physics, we derive a family of associated differential equations that share the same “degenerate” canonical form. These equations can be solved easily if the original equation is known to [...] Read more.
For each fundamental and widely used ordinary second-order linear homogeneous differential equation of mathematical physics, we derive a family of associated differential equations that share the same “degenerate” canonical form. These equations can be solved easily if the original equation is known to possess analytic solutions, otherwise their properties and the properties of their solutions are de facto known as they are comparable to those already deduced for the fundamental equation. We analyze several particular cases of new families related to some of the famous differential equations applied to physical problems, and the degenerate eigenstates of the radial Schrödinger equation for the hydrogen atom in N dimensions. Full article
(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)
23 pages, 3481 KiB  
Article
Ranking Road Sections Based on MCDM Model: New Improved Fuzzy SWARA (IMF SWARA)
by Sabahudin Vrtagić, Edis Softić, Marko Subotić, Željko Stević, Milan Dordevic and Mirza Ponjavic
Axioms 2021, 10(2), 92; https://doi.org/10.3390/axioms10020092 - 15 May 2021
Cited by 54 | Viewed by 3582
Abstract
Traffic management is a significantly difficult and demanding task. It is necessary to know the main parameters of road networks in order to adequately meet traffic management requirements. Through this paper, an integrated fuzzy model for ranking road sections based on four inputs [...] Read more.
Traffic management is a significantly difficult and demanding task. It is necessary to know the main parameters of road networks in order to adequately meet traffic management requirements. Through this paper, an integrated fuzzy model for ranking road sections based on four inputs and four outputs was developed. The goal was to determine the safety degree of the observed road sections by the methodology developed. The greatest contribution of the paper is reflected in the development of the improved fuzzy step-wise weight assessment ratio analysis (IMF SWARA) method and integration with the fuzzy measurement alternatives and ranking according to the compromise solution (fuzzy MARCOS) method. First, the data envelopment analysis (DEA) model was applied, showing that three road sections have a high traffic risk. After that, IMF SWARA was applied to determine the values of the weight coefficients of the criteria, and the fuzzy MARCOS method was used for the final ranking of the sections. The obtained results were verified through a three-phase sensitivity analysis with an emphasis on forming 40 new scenarios in which input values were simulated. The stability of the model was proven in all phases of sensitivity analysis. Full article
(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)
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14 pages, 276 KiB  
Article
Regularization of the Ill-Posed Cauchy Problem for Matrix Factorizations of the Helmholtz Equation on the Plane
by Davron Aslonqulovich Juraev and Samad Noeiaghdam
Axioms 2021, 10(2), 82; https://doi.org/10.3390/axioms10020082 - 2 May 2021
Cited by 16 | Viewed by 2226
Abstract
In this paper, we present an explicit formula for the approximate solution of the Cauchy problem for the matrix factorizations of the Helmholtz equation in a bounded domain on the plane. Our formula for an approximate solution also includes the construction of a [...] Read more.
In this paper, we present an explicit formula for the approximate solution of the Cauchy problem for the matrix factorizations of the Helmholtz equation in a bounded domain on the plane. Our formula for an approximate solution also includes the construction of a family of fundamental solutions for the Helmholtz operator on the plane. This family is parameterized by function K(w) which depends on the space dimension. In this paper, based on the results of previous works, the better results can be obtained by choosing the function K(w). Full article
(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)

Review

Jump to: Editorial, Research

33 pages, 3635 KiB  
Review
Foundations of Engineering Mathematics Applied for Fluid Flows
by Yuli D. Chashechkin
Axioms 2021, 10(4), 286; https://doi.org/10.3390/axioms10040286 - 29 Oct 2021
Cited by 33 | Viewed by 2492
Abstract
Based on a brief historical excursion, a list of principles is formulated which substantiates the choice of axioms and methods for studying nature. The axiomatics of fluid flows are based on conservation laws in the frames of engineering mathematics and technical physics. In [...] Read more.
Based on a brief historical excursion, a list of principles is formulated which substantiates the choice of axioms and methods for studying nature. The axiomatics of fluid flows are based on conservation laws in the frames of engineering mathematics and technical physics. In the theory of fluid flows within the continuous medium model, a key role for the total energy is distinguished. To describe a fluid flow, a system of fundamental equations is chosen, supplemented by the equations of the state for the Gibbs potential and the medium density. The system is supplemented by the physically based initial and boundary conditions and analyzed, taking into account the compatibility condition. The complete solutions constructed describe both the structure and dynamics of non-stationary flows. The classification of structural components, including waves, ligaments, and vortices, is given on the basis of the complete solutions of the linearized system. The results of compatible theoretical and experimental studies are compared for the cases of potential and actual homogeneous and stratified fluid flow past an arbitrarily oriented plate. The importance of studying the transfer and transformation processes of energy components is illustrated by the description of the fine structures of flows formed by a free-falling drop coalescing with a target fluid at rest. Full article
(This article belongs to the Special Issue Modern Problems of Mathematical Physics and Their Applications)
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